Subgroups — #27302345

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810433	C₂	1	4	yes	maximal, complemented
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274879	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133149	C₂²	1	2	yes	—
8	#16810489	—	2	1	no	—
9	#16810433	—	2	8	no	—
10	#16810399	—	2	1	no	—
11	C₄×#11719908	—	2	1	no	—
12	C₂×#135×C₅₃	—	2	1	no	—
13	C₂×#135×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#532	—	2	1	no	—
15	C₅₃×#495	—	2	1	no	—
16	C₅₃×#495	C₄	1	2	yes	complemented
17	C₅₃×#494	C₄	1	1	yes	characteristic, complemented
18	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#492	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	#13133157	—	2	2	no	—
23	#13133157	D₄	1	4	yes	complemented
24	C₂×#11719908	—	2	2	no	—
25	#13133151	—	4	2	no	—
26	#13133149	—	2	4	no	—
27	#13133124	—	4	2	no	—
28	C₂×#41×C₅₃	—	2	2	no	—
29	#135×C₅₃	—	2	4	no	—
30	#135×C₅₃	—	2	4	no	—
31	#135×C₅₃	—	4	2	no	—
32	#135×C₅₃	—	4	2	no	—
33	C₂×Q₁₆×C₅₃	—	2	1	no	—
34	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₅₃	—	2	2	no	—
36	C₂×SD₁₆×C₅₃	—	2	2	no	—
37	C₂×D₈×C₅₃	—	2	1	no	—
38	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	—	2	1	no	—
42	#127×C₅₃	—	2	1	no	—
43	#124×C₅₃	—	2	2	no	—
44	#121×C₅₃	—	4	2	no	—
45	C₄×D₄×C₅₃	—	4	2	no	—
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	#41×C₅₃	—	4	2	no	—
52	#41×C₅₃	—	4	2	no	—
53	#41×C₅₃	—	4	2	no	—
54	#41×C₅₃	—	4	2	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	D₈	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	4	2	no	—
63	Q₁₆×C₅₃	—	2	4	no	—
64	Q₁₆×C₅₃	—	4	2	no	—
65	SD₁₆×C₅₃	—	4	4	no	—
66	SD₁₆×C₅₃	—	4	4	no	—
67	D₈×C₅₃	—	2	4	no	—
68	D₈×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	D₈	1	2	yes	—
72	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₂²⋊C₄×C₅₃	—	4	4	no	—
79	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₅₃	SD₁₆	1	2	yes	—
82	C₄⋊Dic₅₃	—	4	2	no	—
83	C₄⋊Dic₅₃	—	4	2	no	—
84	#11719908	—	2	4	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₅₃	—	4	2	no	—
88	Q₈×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	4	2	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	8	4	no	—
92	D₄×C₅₃	—	8	2	no	—
93	D₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	#102	1	2	yes	—
99	C₂×C₄×C₅₃	#102	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×Dic₅₃	—	8	4	no	—
107	C₂×Dic₅₃	—	8	2	no	—
108	C₈×C₅₃	#102	1	2	yes	—
109	C₈×C₅₃	#102	1	2	yes	—
110	C₈×C₅₃	C₄×D₄	1	2	yes	—
111	C₈×C₅₃	#121	1	2	yes	—
112	#13211	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	8	4	no	—
115	C₂²×C₅₃	—	8	2	no	—
116	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
117	C₂²×C₅₃	#486	1	1	yes	characteristic
118	C₂²×C₅₃	#493	1	1	yes	characteristic
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	8	2	no	—
122	C₄×C₅₃	#390	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
124	C₄×C₅₃	#486	1	1	yes	characteristic
125	C₄×C₅₃	#492	1	1	yes	characteristic
126	Dic₅₃	—	16	2	no	—
127	#3147	D₅₃	1	1	yes	characteristic
128	C₂×#360	—	53	4	no	—
129	C₄×M₃₂	—	53	1	no	—
130	#1846	—	53	1	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	16	2	no	—
134	C₂×C₅₃	#1957	1	1	yes	characteristic
135	C₂×C₅₃	#2145	1	2	yes	—
136	C₂×#135	—	2	1	no	—
137	C₂×#135	C₂×D₅₃	1	2	yes	—
138	#532	—	2	1	no	—
139	C₂×M₃₂	—	53	2	no	—
140	#495	—	2	1	no	—
141	#495	Dic₅₃	1	2	yes	complemented
142	#494	Dic₅₃	1	1	yes	characteristic, complemented
143	#493	Dic₅₃	1	1	yes	characteristic, complemented
144	#492	—	2	1	no	—
145	C₂×#107	—	106	1	no	—
146	C₂×#106	—	53	2	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	#382	—	106	1	no	—
149	#360	—	106	8	no	—
150	#346	—	106	1	no	—
151	C₄×C₁₆	—	106	1	no	—
152	C₅₃	#13211	1	1	yes	characteristic, complemented, Sylow
153	C₂×#41	—	2	2	no	—
154	#135	—	2	4	no	—
155	#135	—	2	4	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	1	no	—
167	#127	—	2	1	no	—
168	#124	—	2	2	no	—
169	#121	—	4	2	no	—
170	C₄×D₄	—	4	2	no	—
171	C₂×C₄⋊C₄	—	106	2	no	—
172	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
173	M₃₂	—	53	4	no	—
174	M₃₂	—	106	2	no	—
175	C₂×C₁₆	—	106	2	no	—
176	#107	—	212	2	no	—
177	#106	—	106	4	no	—
178	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	#95	—	212	2	no	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	212	2	no	—
193	C₂²×C₄	#11719922	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	Q₁₆	—	2	4	no	—
196	Q₁₆	—	4	2	no	—
197	SD₁₆	—	4	4	no	—
198	SD₁₆	—	4	4	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	2	no	—
201	C₂×C₈	#11719922	1	2	yes	—
202	C₂×C₈	#11719923	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	D₄×D₅₃	1	2	yes	—
206	C₂×C₈	#11719947	1	2	yes	—
207	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	212	2	no	—
211	C₄⋊C₄	—	212	2	no	—
212	C₂²⋊C₄	—	4	4	no	—
213	C₄²	#11719923	1	2	yes	—
214	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
216	C₁₆	—	106	4	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₂×#11719922	1	1	yes	characteristic
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	424	4	no	—
231	C₂×C₄	—	424	2	no	—
232	C₂×C₄	#13133138	1	2	yes	—
233	C₂×C₄	#13133138	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
235	C₂×C₄	#13133204	1	1	yes	characteristic
236	C₂×C₄	#13133213	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
240	C₈	#13133138	1	2	yes	—
241	C₈	#13133138	1	2	yes	—
242	C₈	D₄×Dic₅₃	1	2	yes	—
243	C₈	#13133188	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	C₂×#13133138	1	1	yes	characteristic
248	C₂²	#16810969	1	1	yes	characteristic
249	C₂²	#16810999	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	848	2	no	—
254	C₄	#16810526	1	1	yes	characteristic
255	C₄	C₂×#13133138	1	1	yes	characteristic
256	C₄	#16810969	1	1	yes	characteristic
257	C₄	#16810995	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	16	2	no	—
261	C₂	#20275522	1	1	yes	characteristic
262	C₂	#20276451	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3147	C₂
socle	C₂²×C₅₃	C₂×#102
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3147	D₅₃
hypocenter	C₅₃	#13211
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#16810433	C₂	semidirect
2. C₄×#13133157	C₂	semidirect
3. #20274879	C₂	semidirect
16. C₅₃×#495	C₄	semidirect
17. C₅₃×#494	C₄	semidirect
18. C₅₃×#493	C₄	semidirect
23. #13133157	D₄	semidirect
141. #495	Dic₅₃	semidirect
142. #494	Dic₅₃	semidirect
143. #493	Dic₅₃	semidirect
152. C₅₃	#13211	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16810433	2
C₄×#13133157	2
#20274879	2
C₅₃×#3147	2
#13211	53